# Which Test to Use: Ratio or Root? Understanding the Convergence of Series

• MissP.25_5
In summary, when determining whether a series is convergent or divergent, the choice between using the ratio test or root test depends on the form of the terms in the series. Some forms may be more amenable to one test over the other, and experience can also be helpful in making this decision. It is recommended to try multiple tests if unsure.
MissP.25_5
Hello.
How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.

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I would say that whichever test works is the test that you apply it to. It all boils down to the form of the terms in the series that you are interested in -- for certain forms, some tests may not be able to arrive at conclusive results or may be extremely inconvenient.

The first series that you listed down, $\frac{n!^{2}}{(2n)!}$, is clearly amenable to attack by the ratio test since it is straightforward to evaluate $\frac{a_{n+1}}{a_{n}}$. Whereas, for the second series, $\left(\frac{n}{n+1}\right)^{n^2}$, it is not immediately obvious how to evaluate $\frac{a_{n+1}}{a_{n}}$ in a workable form, and hence the ratio test is not easy or convenient to apply to it. In fact, since the terms contain $n$ in the power, this suggests that the root test will be helpful.

Experience of course helps a lot in deciding a lot on which test to use. It wouldn't hurt though to attempt to try several tests (there are definitely some series that can be tackled with multiple tests), if you are not immediately sure which one works.

1 person

## 1. What is the difference between the ratio test and the root test?

The ratio test and root test are both convergence tests used to determine if a series converges or diverges. The main difference between the two is the way in which they compare the terms of the series to its limit. The ratio test compares the ratio of consecutive terms to the limit, while the root test compares the nth root of the absolute value of the nth term to the limit.

## 2. How do you use the ratio test to determine convergence?

To use the ratio test, we take the limit as n approaches infinity of the absolute value of the (n+1)th term divided by the nth term. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and another test must be used.

## 3. How do you use the root test to determine convergence?

To use the root test, we take the limit as n approaches infinity of the nth root of the absolute value of the nth term. If the resulting limit is less than 1, the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, the test is inconclusive and another test must be used.

## 4. Can the ratio test and root test be used on all series?

No, the ratio test and root test are only applicable to series with positive terms. They cannot be used on alternating series or series with negative terms.

## 5. What happens if the limit in the ratio test or root test is equal to 0?

If the limit is equal to 0, the test is inconclusive and another test must be used to determine the convergence or divergence of the series.

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